adding a load bearing wall perpendicular to joists

since i am also renovating the area right below the infamous aberration, i decided to try to assuage the damage somewhat by making two walls that face each other and are 36" apart, load bearing. they are perpendicular to the joists and conveniently right below the sagged part, which i have sistered since the previous post and the joists are considerably more solid (do not sag when i walk on a single joist). one is for the boiler alcove (pictured), the other for the powder room right across the narrow hallway (not yet started).

please consider the photo attached. to make it sturdier, i decided to double up the top plate. so the one on the bottom actually goes under the top plates of the perpendicular wall of the alcove. the top one is 53" wide, the bottom one 60 (53 + 3.5 on each side). what i have not put before taking the picture is the door framing because my question is whether it makes sense to use 4x4 instead of 2x4 for the door framing to make it stronger, which will be in the middle and about 15" from each end approximately. also, does it make sense to use 4x4 instead of the corner studs to make it even stringer. that would give 4 4x4 ove 60", i think that would reinforce the damaged joists above rather firmly. and then i was gonna do the same for the powder room.

then, if you think it makes sense, i can also use those connectors that look like half a hurricane tie, just two plates perpendicular to each other to connect the top plate to the joist above.